Thermodynamic Parametrisation of the Vertebrate Lifetime Cycle Invariant: Biological Proper Time, Allometric Mass-Cancellation, and Clade-Specific Predictions

Abstract

Warm-blooded vertebrates accumulate approximately $\Nstar \approx 10^9$ cardiac cycles over a natural lifetime, a striking empirical regularity first quantified by Lindstedt and Calder yet lacking a physical interpretation. We propose that this invariance is consistent with a conserved thermodynamic budget, formulated here as the Principle of Biological Time Equivalence (PBTE). The framework rests on a constitutive closure $\dot{\Sigma} = \sigma_0 f$, which links the entropy production rate to the intrinsic physiological frequency; integration over the lifespan yields $\Sigma_{\mathrm{life}} = \sigma_0 \Nstar$, so that the observed constancy of $\Nstar$ corresponds to an approximately constant lifetime entropy budget. Algebraic exponent cancellation under Kleiber and Calder scaling laws, $\sigstar \propto M^{3/4+1/4-1}=M^0$, is consistent with mass-independence and reproduces the numerical value $N_0 \approx 1.52\times10^9$ without free parameters. The framework offers a thermodynamically consistent account of two outstanding problems: the origin of the numerical value of $\Nstar$ and the systematic deviations observed across clades. A multiplicative correction factor $\Phi_C$, constructed from physiological determinants -- activity allocation, body temperature, mitochondrial efficiency, and extrinsic hazard -- predicts long-lived clades as regimes of reduced effective entropy production per cardiac cycle.